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Welcome to the site. Those were both great and welcome questions; I hope you don't mind; but I was hoping you could ask your second question : "For a person with mathematical background what would be a good way to dive into the field of cognitive science?" as a separate question; the site works best when there is one topic per question (unless they are highly related, which in this case, I don't think they are).
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Jeromy Anglim♦Feb 6 '12 at 0:02

4 Answers
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Given your background and interest in modeling, I would highly recommend The Cambridge Handbook of Computational Psychology. The book provides an overview for several of the prominent modeling paradigms in cogsci, including dynamical systems, as well as many concrete examples--albeit most using other computational paradigms.

Dynamical systems, to my knowledge, are most used in the field of motor control. Though to be honest, dyn systems is not 'my' paradigm, and most of my knowledge comes from the aforementioned book.

A highly illustrative example comes from the orientation behaviors of
the common house ﬂy (Reichardt & Poggio, 1976; Poggio & Reichardt,
1976). Flies orient toward moving objects, which they chase as part of
their mating behavior. Detailed analysis revealed that the circuitry
underlying this behavior forms a simple controller: a motion detection
system fed by luminance changes on the ﬂy’s facet eye drives the ﬂight
motor, generating an amount of torque that is a function of where on
the sensory surface motion was detected. If the speck of motion is
detected on the right, a torque to the right is generated. If the
speck is detected on the left, a torque to the left is generated. The
level of torque passes through zero when the speck is right ahead. The
torque changes the ﬂight direction of the ﬂy, which in turn changes
the location on the facet eye at which the moving stimulus is
detected. Given the aerodynamics of ﬂies, the torque and its on-line
updating generate an orientation behavior, in which the insect orients
its ﬂight into the direction in which a moving stimulus is detected.

I have a similar background to you, and have found a lot of interesting things in evolutionary game theory (you can follow links from my profile for more). But on the specific content of your question: I have come across to uses of dynamic systems on the opposite ends of cognition. Beer's work on modeling minimal cognition, and Busemeyer & Townsend's work on human decision making.

Minimal cognition

Beer's work models simple cognitive agents as dynamic recurrent neural networks. He then either hand-builds the agents, or evolves them, and analyzes them from a situated cognition perspective by treating them as dynamic systems.

Human decision making

Busemeyer & Towsend on the other hand try to describe how humans deliberate and make decisions. One of the key aspects they want to capture is not just the outcome of human decision making (with all of its flaws and irrationality) but also the deliberation time. To do this, they introduced decision field theory which models human decision making as a stochastic dynamic process.

Is Busemeyer's work really anything beyond application of decades-old models of perceptual decision making (diffusion, various accumulator models) to more abstract non-perceptual judgements?
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Mike LawrenceFeb 6 '12 at 2:44

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@Mike I don't know enough about the decades-old models of perceptual decision making to comment. I don't think the authors viewed their work as just routine application though. They obviously use diffusion equations... but deeming a work as a basic application only on account of that would make almost everything a basic application of statistical mechanics (which might be a true judgement ;)).
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Artem Kaznatcheev♦Feb 6 '12 at 2:51